Theorem dcomex 10026

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	⊢ ω ∈ V

Proof of Theorem dcomex

Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8199	. . . . . . 7 ⊢ 1o ≠ ∅
2		df-br 5040	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3		elsni 4544	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4		fvex 6708	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6708	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 5344	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓‘𝑛) = 1o)
7	3, 6	syl 17	. . . . . . . 8 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓‘𝑛) = 1o)
8	2, 7	sylbi 220	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1o)
9		tz6.12i 6721	. . . . . . 7 ⊢ (1o ≠ ∅ → ((𝑓‘𝑛) = 1o → 𝑛𝑓1o))
10	1, 8, 9	mpsyl 68	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11		vex 3402	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1oex 8193	. . . . . . 7 ⊢ 1o ∈ V
13	11, 12	breldm 5762	. . . . . 6 ⊢ (𝑛𝑓1o → 𝑛 ∈ dom 𝑓)
14	10, 13	syl 17	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
15	14	ralimi 3073	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16		dfss3 3875	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17	15, 16	sylibr 237	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18		vex 3402	. . . . 5 ⊢ 𝑓 ∈ V
19	18	dmex 7667	. . . 4 ⊢ dom 𝑓 ∈ V
20	19	ssex 5199	. . 3 ⊢ (ω ⊆ dom 𝑓 → ω ∈ V)
21	17, 20	syl 17	. 2 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22		snex 5309	. . 3 ⊢ {⟨1o, 1o⟩} ∈ V
23	12, 12	fvsn 6974	. . . . . . . 8 ⊢ ({⟨1o, 1o⟩}‘1o) = 1o
24	12, 12	funsn 6411	. . . . . . . . 9 ⊢ Fun {⟨1o, 1o⟩}
25	12	snid 4563	. . . . . . . . . 10 ⊢ 1o ∈ {1o}
26	12	dmsnop 6059	. . . . . . . . . 10 ⊢ dom {⟨1o, 1o⟩} = {1o}
27	25, 26	eleqtrri 2830	. . . . . . . . 9 ⊢ 1o ∈ dom {⟨1o, 1o⟩}
28		funbrfvb 6745	. . . . . . . . 9 ⊢ ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
29	24, 27, 28	mp2an 692	. . . . . . . 8 ⊢ (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
30	23, 29	mpbi 233	. . . . . . 7 ⊢ 1o{⟨1o, 1o⟩}1o
31		breq12 5044	. . . . . . . 8 ⊢ ((𝑠 = 1o ∧ 𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
32	12, 12, 31	spc2ev 3512	. . . . . . 7 ⊢ (1o{⟨1o, 1o⟩}1o → ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡)
33	30, 32	ax-mp 5	. . . . . 6 ⊢ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡
34		breq 5041	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1o, 1o⟩}𝑡))
35	34	2exbidv 1932	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡))
36	33, 35	mpbiri 261	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
37		ssid 3909	. . . . . . 7 ⊢ {1o} ⊆ {1o}
38	12	rnsnop 6067	. . . . . . 7 ⊢ ran {⟨1o, 1o⟩} = {1o}
39	37, 38, 26	3sstr4i 3930	. . . . . 6 ⊢ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40		rneq 5790	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41		dmeq 5757	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
42	40, 41	sseq12d 3920	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
43	39, 42	mpbiri 261	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44		pm5.5 365	. . . . 5 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
45	36, 43, 44	syl2anc 587	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
46		breq 5041	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
47	46	ralbidv 3108	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
48	47	exbidv 1929	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
49	45, 48	bitrd 282	. . 3 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50		ax-dc 10025	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
51	22, 49, 50	vtocl 3464	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
52	21, 51	exlimiiv 1939	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2112   ≠ wne 2932  ∀wral 3051  Vcvv 3398   ⊆ wss 3853  ∅c0 4223  {csn 4527  ⟨cop 4533   class class class wbr 5039  dom cdm 5536  ran crn 5537  suc csuc 6193  Fun wfun 6352  ‘cfv 6358  ωcom 7622  1_oc1o 8173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501  ax-dc 10025

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-fv 6366  df-1o 8180

This theorem is referenced by:  axdc2lem  10027  axdc3lem  10029  axdc4lem  10034  axcclem  10036